Problem: $ A = \left[\begin{array}{rr}-2 & 0 \\ 0 & 2 \\ 0 & 1\end{array}\right]$ $ F = \left[\begin{array}{rr}-1 & 4 \\ -1 & 1 \\ 1 & -2\end{array}\right]$ Is $ A+ F$ defined?
Solution: In order for addition of two matrices to be defined, the matrices must have the same dimensions. If $ A$ is of dimension $( m \times  n)$ and $ F$ is of dimension $( p \times  q)$ , then for their sum to be defined: 1. $ m$ (number of rows in $ A$ ) must equal $ p$ (number of rows in $ F$ ) and 2. $ n$ (number of columns in $ A$ ) must equal $ q$ (number of columns in $ F$ Do $ A$ and $ F$ have the same number of rows? Yes Yes No Yes Do $ A$ and $ F$ have the same number of columns? Yes Yes No Yes Since $ A$ has the same dimensions $(3\times2)$ as $ F$ $(3\times2)$, $ A+ F$ is defined.